Also have a look at the ITIL roles and seeing who you can do these roles in an organisation. E.g. Change Manager is a role and would get you great exposure and experience if you can do that - incident manager or service desk team lead / supervisor etc are all roles you can do to gain experience.

Even in networking you can look at doing some form of problem management. Don't expect to do the practioners and thats all you need. You do need relevant experience also. Both go hand in hand._________________Mark O'Loughlin
ITSM / ITIL Consultant

I think that helping your company implement capacity & availability would be a +ve that follows your current technical career path. Work on some Proactive Problems, help with the budgeting process (Financial Mgt).

Chances are that your company tries to get SD, Incident, Problem & Change under control first.

Maybe take a look at Practioner training for some of the Delivery processes.